To start with, while I know you're asking specifically about man-portable gunpowder weapons, it must still be said that most gunpowder weapons in the Middle Ages were artillery. Pretty much up until the invention of the match-lock during the fifteenth century artillery dominated gunpowder technology. That means that while early gunpowder weapons may have outperformed bows and crossbows in some areas, their sparse presence in medieval warfare meant that they didn't completely eclipse those weapons. Also, for the sake of simplicity I'm just going to use 'guns' for the rest of this response, but assume I mean the arquebus when I say it. The exact terminology of medieval gunpowder weaponry is a deep rabbit hole of debate that I'm just staying outside of for this answer.

In general, guns outperformed crossbows (and bows) when it came to power. Thom Richardson it an interesting set of tests conducted at the Royal Armouries in the UK showed that early gunpowder weapons (in this case roughly 15th century) shot their bullets with a force an order of magnitude higher than either the bow or the crossbow could. While these tests are not perfect since they used a modern blackpowder, which is only really equivalent to 16th century or later gunpowder, and one could argue their crossbow was a tad on the weak side the overall results are significant enough to be convincing. Early guns were much more capable of penetrating armour and shot their projectiles much faster than anything else available at the time.

That's a pretty clear advantage, but guns were not without their disadvantages. Logistics in particular was a real problem. Before the invention of corning in either the late 14th or early 15th century, gunpowder was very prone to being ruined by damp conditions. It also had to be prepared on the spot because the three main ingredients, when transported, would often fail to remain mixed together, and would instead separate into their component parts, which would still ignite when lit but wouldn't trigger a significant explosion. After the invention of corning, which was essentially a process where the ingredients were mixed together with a liquid and then dried into a cake which was then crumbled, there was still the problem of cost. Saltpeter remained very expensive up until the late 15th century, so gunpowder weapons were much more expensive to operate than the bow or the crossbow. Lastly, despite their reputation for being a weapon where 'any peasant could kill a knight' guns were actually very difficult to use well and would have required specialist training. Before the match-lock there was no point and shoot interface for using a gun, and guns had to be manually loaded, there were no cartridges. The earliest gunners even had to mix their own gunpowder, making sure they got the ratio of ingredients correct each time. Archers and crossbowmen were also often specialists, or at least semi-professional soldiers, by the Later Middle Ages so this last part wouldn't necessarily mean a higher cost, it certainly didn't offer a money saving alternative either.

There also remain a few other things to consider when it came to early gunpowder weaponry. Firstly, and perhaps obviously, there was no smokeless powder so anyone employing early guns ran the risk of obscuring the battlefield and creating confusion, which wasn't always necessarily a benefit. Gunpowder weapons were also very slow to reload. While a heavy crossbow was also very slow to reload, lighter crossbows could be reloaded relatively quickly. Accuracy would also be a problem. There's some evidence for rifling from the very end of the fifteenth century, but even if that is actual rifling it was certainly extraordinarily rare during this time. While in theory a smooth bore weapon can be quite accurate if the bullet fits the barrel snugly, making custom bullets to fit each of your guns and employing master craftsmen to do so would represent a significant expense. Not an impossible one, mind, but a serious burden on any lord wishing to employ guns that had a significant level of accuracy.

Esper, Thomas, ‘The Replacement of the Longbow by Firearms in English Army’, Technology and Culture, Vol. 6 No. 3 (1965). Pp. 382-393.

Richardson, Thom, ‘Ballistic Testing of Historical Weapons’, Royal Armouries Yearbook, Vol. 3 (1998). Pp. 50-52.

Williams, Alan, ‘Some Firing Tests with Simulated Fifteenth-Century Handguns’, The Journal of the Arms and Armour Society, Vol. 8 No. 1, (1974). pp. 114-120.

DeVries, Kelly, 'Gunpowder and Early Gunpowder Weapons', Kelly DeVries (ed.), Guns and Men in Medieval Europe, 1200-1500, (Aldershot, 2002). pp. 121-135.

Claude Blair (ed.), Pollard's History of Firearms, (Feltham, 1983).

Tout, T.F., ‘Firearms in England in the Fourteenth Century’, The English Historical Review, Vol. 26 No. 104 (1911). pp. 666-702.

You may also be interested in my answer to a similar question from a little while ago

before sophisticated gunpowder and the other things that lead to the Hook Gun, Medieval firearms had a limited battlefield presence. They were inaccurate and lacked power. They were relatively easy to use and easy to mass produce (compared to bows) but crossbows would generally have greater accuracy and in some cases better armor penetration (this would depend on a wide variety of factors. range, ammunition used, and so forth.) These early guns were relatively large and generally could not be fired from the shoulder. I think the fact that Crossbows were common and Guns were not speaks for itself. they filled a different battlefield roll, but the Crossbow had a few advantages that helped it remain dominant for a very long time. I would not consider Guns in the late middle ages a step up from the Crossbow.